Magnetic sensor with an asymmetric wheatstone bridge

ABSTRACT

Magnetic sensors, sensor modules, and methods thereof are provided. A magnetic sensor includes a sensor arrangement including a plurality of magnetic field sensor elements electrically arranged in an asymmetrical bridge circuit, where a first total resistance of a first pair of sensor elements provided on a first side of the asymmetrical bridge circuit is different from a second total resistance of a second pair of sensor elements provided on a second side of the asymmetrical bridge circuit, and the asymmetrical bridge circuit is configured to generate a differential signal based on sensor signals generated by the plurality of magnetic field sensor elements in response to a magnetic field impinging thereon.

FIELD

The present disclosure relates generally magnetic sensors, and, moreparticularly, to a magnetic sensor with an asymmetric Wheatstone bridge.

BACKGROUND

In the field of speed sensing, a sinusoidal-like signal may be generatedby a magnetic sensor in response to a rotation of a target object, suchas a wheel, camshaft, crankshaft, or the like. For example, a back biasmagnet may be used to produce a magnetic field, and a magnetic speedsensor may be placed between the target object and the back bias magnet.The sinusoidal signal may be generated and translated into pulses, whichis further translated into a movement detection or a speed output.

One or more sensor arrangements may be used in a Wheatstone bridgeconfiguration. However, in back bias applications, when a magneticsensor is placed between a toothed wheel and a magnet, the magneticfield of a standard low cost magnet generates a large static magneticfield to the left side and the right side of the bridge. This largestatic magnetic field results in a large electrical direct current (DC)offset at the output voltage of the Wheatstone bridge. The electrical DCoffset must be compensated by an offset compensation digital-to-analogconverter (DAC) or similar solution. A drawback using the offsetcompensation DAC is that additional noise is introduced into the signalpath. This is especially the case if a ratio between the offset and thesignal is large, as is usually the case. Then, the offset compensationDAC must be optimized for noise which requires very large chip area.

SUMMARY

Magnetic sensors, sensor modules, and methods thereof are provided.

According to one or more embodiments, a magnetic sensor includes asensor arrangement including a plurality of magnetic field sensorelements electrically arranged in an asymmetrical bridge circuit, wherea first total resistance of a first pair of sensor elements provided ona first side of the asymmetrical bridge circuit is different from asecond total resistance of a second pair of sensor elements provided ona second side of the asymmetrical bridge circuit, and the asymmetricalbridge circuit is configured to generate a differential signal based onsensor signals generated by the plurality of magnetic field sensorelements in response to a magnetic field impinging thereon.

According to one or more embodiments, a magnetic sensor module includesa magnet having a magnetic operation point (MOP) and configured toproduce a differential magnetic field, the differential magnetic fieldhaving a differential MOP field strength at the MOP, and further havinga first differential field portion and a second differential fieldportion. The magnetic sensor module further includes an asymmetricbridge circuit including a plurality of magnetic field sensor elements,including a first pair of sensor elements disposed in the firstdifferential field portion and a second pair of sensor elements disposedin the second differential field portion, where the asymmetric bridgecircuit is configured to generate a differential signal based on sensorsignals generated by the plurality of magnetic field sensor elements,and the differential signal is zero on a condition that a differentialfield strength of the differential magnetic field impinging on theplurality of magnetic field sensor elements is equivalent to thedifferential MOP field strength.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments are described herein making reference to the appendeddrawings.

FIGS. 1A and 1B shows a cross-sectional view illustrating a magneticfield sensing principle according to one or more embodiments;

FIG. 1C illustrates a magnetic field sensing principle with a magneticencoder according to one or more embodiments;

FIG. 2 is an example of a normalized sinusoidal waveform generated by asensor arrangement of a magnetic speed sensor;

FIG. 3 is a schematic block diagram illustrating a magnetic speed sensoraccording to one or more embodiments;

FIG. 4 is a schematic diagram illustrating two example sensor bridgeconfigurations using four xMR sensor elements according to one or moreembodiments;

FIGS. 5A-5C show plan views of different example sensor arrangementconfigurations of a magnetic sensor according to one or moreembodiments; and

FIG. 6 shows a schematic diagram illustrating an asymmetric sensorbridge circuit according to one or more embodiments.

DETAILED DESCRIPTION

In the following, various embodiments will be described in detailreferring to the attached drawings. These embodiments are given forillustrative purposes only and are not to be construed as limiting. Forexample, while embodiments may be described as comprising a plurality offeatures or elements, in other embodiments, some of these features orelements may be omitted, and/or may be replaced by alternative featuresor elements. In other embodiments, further features or elements inaddition to those explicitly shown or described may be provided. Inaddition, features of the different embodiments described hereinaftermay be combined with each other to form further embodiments, unlessspecifically noted otherwise. For example, variations or modificationsdescribed with respect to one of the embodiments may also be applicableto other embodiments unless noted to the contrary.

Accordingly, while further examples are capable of various modificationsand alternative forms, some particular examples thereof are shown in thefigures and will subsequently be described in detail. However, thisdetailed description does not limit further examples to the particularforms described. Further examples may cover all modifications,equivalents, and alternatives falling within the scope of thedisclosure.

Further, equivalent or like elements or elements with equivalent or likefunctionality are denoted in the following description with equivalentor like reference numerals. As the same or functionally equivalentelements are given the same reference numbers in the figures, a repeateddescription for elements provided with the same reference numbers may beomitted. Hence, descriptions provided for elements having the same orlike reference numbers are mutually exchangeable.

Whenever a singular form such as “a,” “an,” and “the” is used and usingonly a single element is neither explicitly or implicitly defined asbeing mandatory, further examples may also use plural elements toimplement the same functionality. Likewise, when a functionality issubsequently described as being implemented using multiple elements,further examples may implement the same functionality using a singleelement or processing entity. It will be further understood that theterms “comprises,” “comprising,” “includes,” and/or “including,” whenused, specify the presence of the stated features, integers, steps,operations, processes, acts, elements and/or components, but do notpreclude the presence or addition of one or more other features,integers, steps, operations, processes, acts, elements, componentsand/or any group thereof.

It will be understood that when an element is referred to as being“connected” or “coupled” to another element, it can be directlyconnected or coupled to the other element or intervening elements may bepresent. In contrast, when an element is referred to as being “directlyconnected” or “directly coupled” to another element, there are nointervening elements present. Other words used to describe therelationship between elements should be interpreted in a like fashion(e.g., “between” versus “directly between,” “adjacent” versus “directlyadjacent,” etc.).

In embodiments described herein or shown in the drawings, any directelectrical connection or coupling, i.e., any connection or couplingwithout additional intervening elements, may also be implemented by anindirect connection or coupling, i.e., a connection or coupling with oneor more additional intervening elements, or vice versa, as long as thegeneral purpose of the connection or coupling, for example, to transmita certain kind of signal or to transmit a certain kind of information,is essentially maintained.

The drawings are to be regarded as being schematic representations andelements illustrated in the drawings are not necessarily shown to scale.Rather, the various elements are represented such that their functionand general purpose become apparent to a person skilled in the art. Anyconnection or coupling between functional blocks, devices, components,or other physical or functional units shown in the drawings or describedherein may also be implemented by an indirect connection or coupling.Functional blocks may be implemented in hardware, firmware, software, ora combination thereof.

Embodiments relate to sensors and sensor systems, and to obtaininginformation about sensors and sensor systems. A sensor may refer to acomponent which converts a physical quantity to be measured to anelectric signal, for example, a current signal or a voltage signal. Thephysical quantity may for example comprise a magnetic field, an electricfield, a pressure, a force, a current or a voltage, but is not limitedthereto. A sensor device, as described herein, may be a an angle sensor,a linear position sensor, a speed sensor, motion sensor, and the like.

A magnetic field sensor, for example, includes one or more magneticfield sensor elements that measure one or more characteristics of amagnetic field (e.g., an amount of magnetic field flux density, a fieldstrength, a field angle, a field direction, a field orientation, etc.).The magnetic field may be produced by a magnet, a current-carryingconductor (e.g., a wire), the Earth, or other magnetic field source.Each magnetic field sensor element is configured to generate a sensorsignal (e.g., a voltage signal) in response to one or more magneticfields impinging on the sensor element. Thus, a sensor signal isindicative of the magnitude and/or the orientation of the magnetic fieldimpinging on the sensor element.

It will be appreciated that the terms “sensor”, “sensor element”, and“sensing element” may be used interchangeably throughout thisdescription, and the terms “sensor signal” and “measurement signal” mayalso be used interchangeably throughout this description.

Magnetic sensors, as provided herein, may be magnetoresistive sensors.Magnetoresistance is a property of a material to change the value of itselectrical resistance when an external magnetic field is applied to it.Some examples of magnetoresistive effects are Giant Magneto-Resistance(GMR), which is a quantum mechanical magnetoresistance effect observedin thin-film structures composed of alternating ferromagnetic andnon-magnetic conductive layers, Tunnel Magneto-Resistance (TMR), whichis a magnetoresistive effect that occurs in a magnetic tunnel junction(MTJ), which is a component consisting of two ferromagnets separated bya thin insulator, or Anisotropic Magneto-Resistance (AMR), which is aproperty of a material in which a dependence of electrical resistance onthe angle between the direction of electric current and direction ofmagnetization is observed. For example, in the case of AMR sensors, aresistance for an AMR sensor element changes according to a square of asine of an angle of the magnetic field component projected on a sensingaxis of the ARM sensor element.

The plurality of different magnetoresistive effects is commonlyabbreviated as xMR, wherein the “x” acts as a placeholder for thevarious magnetoresistive effects. xMR sensors can detect the orientationof an applied magnetic field by measuring sine and cosine anglecomponents with monolithically integrated magnetoresistive sensorelements.

Magnetoresistive sensor elements of such xMR sensors typically include aplurality of layers, of which at least one layer is a reference layerwith a reference magnetization (i.e., a reference direction). Thereference magnetization is a magnetization direction that provides asensing direction corresponding to a sensing axis of the xMR sensor.Accordingly, if a magnetic field component points exactly in the samedirection as the reference direction, a resistance of the xMR sensorelement is at a maximum, and, if a magnetic field component pointsexactly in the opposite direction as the reference direction, theresistance of the xMR sensor element is at a minimum. A magnetic fieldcomponent may be, for example, an x-magnetic field component (Bx), ay-magnetic field component (By), or a z-magnetic field component (Bz),where the Bx and By field components are in-plane to the magneticsensor, and Bz is out-of-plane to the magnetic sensor.

In some applications, an xMR sensor includes a plurality ofmagnetoresistive sensor elements, which have different referencemagnetizations. Examples of such applications, in which variousreference magnetizations are used, are angle sensors, compass sensors,or specific types of speed sensors (e.g., speed sensors in a bridgearrangement).

By way of example, such magnetoresistive sensor elements are used inspeed, angle or rotational speed measuring apparatuses, in which magnetsmay be moved relative to an magnetoresistive sensor elements and hencethe magnetic field at the location of the magnetoresistive sensorelement changes in the case of movement, which, in turn, leads to ameasurable change in resistance.

According to one or more embodiments, a magnetic field sensor and asensor circuit may be both accommodated (i.e., integrated) in the samechip package (e.g., a plastic encapsulated package, such as leadedpackage or leadless package, or a surface mounted device (SMD)-package).This chip package may also be referred to as sensor package. The sensorpackage may be combined with a back bias magnet to form a sensor module,sensor device, or the like.

The sensor circuit may be referred to as a signal processing circuitand/or a signal conditioning circuit that receives one or more signals(i.e., sensor signals) from one or more magnetic field sensor elementsin the form of raw measurement data and derives, from the sensor signal,a measurement signal that represents the magnetic field. Signalconditioning, as used herein, refers to manipulating an analog signal insuch a way that the signal meets the requirements of a next stage forfurther processing. Signal conditioning may include converting fromanalog to digital (e.g., via an analog-to-digital converter),amplification, filtering, converting, biasing, range matching, isolationand any other processes required to make a sensor output suitable forprocessing after conditioning.

Thus, the sensor circuit may include an analog-to-digital converter(ADC) that converts the analog signal from the one or more sensorelements to a digital signal. The sensor circuit may also include adigital signal processor (DSP) that performs some processing on thedigital signal, to be discussed below. Therefore, the sensor package mayinclude a circuit that conditions and amplifies the small signal of themagnetic field sensor element via signal processing and/or conditioning.

A sensor device, as used herein, may refer to a device which includes asensor and sensor circuit as described above. A sensor device may beintegrated on a single semiconductor die (e.g., silicon die or chip),although, in other embodiments, a plurality of dies may be used forimplementing a sensor device. Thus, the sensor and the sensor circuitare disposed on either the same semiconductor die or on multiple dies inthe same package. For example, the sensor might be on one die and thesensor circuit on another die such that they are electrically connectedto each other within the package. In this case, the dies may becomprised of the same or different semiconductor materials, such as GaAsand Si, or the sensor might be sputtered to a ceramic or glass platelet,which is not a semiconductor.

Embodiments herein may employ one or more Wheatstone bridge circuits. Asused herein, “bridge circuit,” “bridge device,” and “bridge” may be usedinterchangeably. A bridge circuit is a topology of electrical circuit inwhich two circuit branches or legs (usually in parallel with each other)are “bridged” by a third branch connected between the first two branchesat some intermediate point along them. A bridge circuit may includepassive elements, such as resistive, capacitive, and inductive elements,or a combination thereof.

Magnetoresistive sensor elements of an xMR sensor are resistive elementsthat may be arranged in a bridge configuration. This bridge circuit ofxMR sensors may convert a differential magnetic field applied to a leftside and a right side of the bridge circuit into a differential outputvoltage. Assuming an ideal xMR sensor, the output voltage is linearlyproportional to the differential magnetic field.

In particular, a resistive value of the one or more magnetic fieldsensor elements may change when exposed to a magnetic field. Theresistive value, which corresponds to a characteristic of the magneticfield, may be measured such that information regarding thecharacteristic of the magnetic field is obtained. Further, the resistivevalue may be measured in the form of a voltage or current measurement.Thus, magnetic field sensor elements in an xMR sensor may be arranged ina bridge formation to provide a resistance value (e.g., in the form of avoltage output) corresponding to a measured characteristic of a magneticfield.

Magnetic field sensors provided herein may be configured for incrementalspeed measurement, angle measurement, position measurement, and rotationdirection measurement of a rotating target object, such as a magneticencoder (e.g., wheel), camshaft, or shaft. However, the embodiments arenot limited thereto, and may apply to any device or system which usesmagnetic sensors in a bridge configuration to generate a differentialsensor (measurement) signal.

FIGS. 1A and 1B show a cross-sectional view illustrating a magneticfield sensing principle according to one or more embodiments. A sensordevice 4, which may also be referred to as a sensor chip or a sensorcircuit, may include two or more sensor elements disposed in adifferential magnetic field produced by a magnet 6. The magnet 6 may bea standard low cost permanent magnet that generates a static magneticfield to the left side and the right side of the center of the magnet.This may also be referred to as a differential magnetic field, such thatthe amplitude of the magnetic field at the left side is equal to theamplitude of the magnetic field at the right side.

The magnet 6 generates a static magnetic field, as shown in FIG. 1B. Ifa sensor (e.g., sensor element 5L or 5R) is placed at a certain distancefrom the magnet 6, then a static magnetic field is present at the sensorposition. This static magnetic field can be expressed by a component inthe x- and y-direction as illustrated in FIG. 1B. For a magnetic sensingmodule (i.e., including the magnet 6 and the sensor device 4) themagnetic operation point (MOP) is defined as the measured field by thesensor in the absence of a target object, such as a wheel.

If two x-sensitive sensors are used with a certain displacement aboutthe center of the magnet 6, then the MOP of the left sensor 5L and theright sensor 5R have a different sign. The differential field, alsocalled differential MOP, is large. In the presence of a wheel the MOP ofthe sensor is influenced and depends on the angle of the wheel. Thedifferential sensing module with a large MOP (x-sensitive sensors)generates a differential output signal around this MOP, i.e., with alarge offset

Thus, two sensor elements 5L and 5R are shown that are differentiallydisplaced from each other such that one sensor element resides in halfof the differential magnetic field, and the other sensor element residesin the other half of the differential magnetic field. The fieldamplitude of the differential magnetic field at the two locations is thesame, but the sign of the magnetic field is opposite. In this example,an x-field component (Bxl) of the magnetic field at sensor element 5L is−20 mT, while the x-field component (Bxr) of the magnetic field atsensor element 5R is +20 mT. The differential field measured is 40 mT.It will be appreciated that field of 20 mT serves only as an example,and that the embodiments provided herein are not limited thereto. Inaddition, sensor elements 5L and 5R may be placed at locations where thefield amplitude is different from each other (e.g., at −20 mT and −10mT).

It will also be appreciated that sensor elements 5L and 5R may eachrepresent a pair of sensor elements where a pair of sensor elements isarranged substantially at each location or region on the sensor chip. Inthis example, the sensor elements 5L and 5R may have opposing referencedirections aligned along a same sensing axis (e.g., the x-axis). Thus,the sensor elements 5L and 5R are sensitive to an x-field component (Bx)produced by the magnet 6. Here, to sense an equivalent x-field component(Bx), sensor elements of a sensor pair may be aligned with each other inthe y-direction. In addition, the reference directions are not limitedto the above orientation and may be fixed in other orientations.

FIG. 1C illustrates a magnetic field sensing principle with a magneticencoder according to one or more embodiments. One type of magneticencoder may be a ferromagnetic encoder, which may be a toothed wheel ora toothed disc of ferromagnetic material with holes or notches that passin front of the magnetic field sensor. Similar to that of FIG. 1A, adifferential magnetic field may be produced by a back bias magnet 6coupled to a backside of the magnetic field sensor. Thus, the magneticfield pattern of the magnetic field produced by the back bias magnet 6may be altered by the passing of teeth 2 and notches 3 of the rotatingmagnetic encoder 1. Hence, the strength of the of the magnetic fieldproduced by the back bias magnet 6 in certain sensing planes and sensingdirections (e.g., x-, y-, and z-planes and directions) may be alteredaccording to the change in the magnetic field pattern.

FIG. 1C shows a toothed wheel 1 that may rotate in either rotationdirection and has alternating teeth 2 and notches 3, according to one ormore embodiments. In particular, the toothed wheel 1 may be made of aferromagnetic material (e.g., iron) that attracts magnetic fields. Inaddition, a sensor device 4 may include two pairs of sensor elements 5Land 5R that are arranged in a bridge configuration and are configured tosense a differential magnetic field produced by a back bias magnet 6.Together, the sensor device 4 and the back bias magnet 6 may comprise asensor module. The sensor device 4 may generally be referred to hereinas sensor 4, may further include a sensor circuit (not shown), and maybe disposed in a sensor package.

As the toothed wheel 1 rotates, the teeth 2 and notches 3 alternatepassing by the sensor module including the back bias magnet 6 and thepair of sensor elements 5L and 5R. In the instance of a tooth 2 passingthe sensor module, the magnetic field lines of the bias magnetic fieldproduced by the back bias magnet 6 are pulled in the z-direction towardsthe tooth 2. Thus, the magnetic field lines are pulled away from the x-and y-planes and a sensed magnetic field strength in the x- andy-directions is reduced such that a minimum field strength of Bx and Byin the in the x- and y-directions would be detected at the center of thetooth 2. In contrast, a sensed magnetic field strength of Bz in thez-direction is increased such that a maximum field strength in thez-direction would be detected at the center of the tooth 2. This maydiffer in real-world applications where the minimum may not occurexactly at the center due to assembly tolerances, but the minimum fieldstrength should be detected substantially at the center of the tooth 2.

Conversely, in the instance of a notch 3 passing the sensor module,magnetic field lines of the bias magnetic field produced by the backbias magnet 6 are not pulled (or are less pulled) in the z-directiontowards the notch 3. Thus, the magnetic field lines remain concentratedrelative to the x- and y-planes and the sensed magnetic field strengthof Bx and By in the x- and y-directions would be at a maximum in the x-and y-directions at the center of the notch 3. In contrast, a sensedmagnetic field strength in the z-direction is reduced such that aminimum field strength in the z-direction would be detected at thecenter of the notch 2. This may differ in real-world applications wherethe maximum may not occur exactly at the center, but the maximum fieldstrength should be detected substantially at the center of the notch 3.

The two pairs of magnetic field sensor elements 5L and 5R in may bereferred to as differential pairs of sensor elements that are configuredto convert the differential magnetic field applied to the left side andthe right side of the sensor device 4 into a differential output voltage(i.e., a differential signal). In such a differential bridgeconfiguration, the sensor signals from each sensor element of thedifferential pair of sensor elements 5L and 5R may be provided withinthe sensor circuit (i.e., the bridge circuit) that is configured togenerate a differential signal at the bridge output. Due to the bridgeconfiguration, the differential signal may cancel out homogeneous and/ornon-homogenous stray-fields in the sensing axis of the xMR sensor.

In addition, each differential pair of sensor elements 5L and 5R may bedisposed from a center of the magnetic field at a distance of about halfof a pitch of wheel 1 in order to generate a differential signal withhigh signal to noise ratio. That is, the distance between thedifferential pair of sensor elements 5L and 5R, both being spaced abouthalf a pitch from center, may be matched or substantially matched (e.g.,within 5% to allow for manufacturing tolerances) to the pitch of thewheel 1. A pitch is the distance along a pitch circle between twoadjacent teeth of a toothed wheel. However, it will be appreciated thatother spacing arrangements are also possible and used especially if thewheel is used to transmit mechanical forces. Lastly, leads 7 provide anelectrical pathway for various input and output signals (e.g., power,command and output signals) to and from the sensor device 4.

Referring to the configuration shown in FIG. 1C as an example, as thewheel 1 rotates, the teeth 2 and notches 3 alternate past the sensormodule and the differential pair of sensor elements (or the fourresistors within a Wheatstone bridge configuration) 5L and 5R sense achange in the x-axis magnetic field strength Bx that varies as asinusoidal-like waveform (i.e., as a signal modulation), the frequencyof which corresponds to a speed of rotation of the wheel 1. Thus, thesensor circuit of the sensor device 4 receives signals (i.e., sensorsignals) from each sensor element of the differential pairs of sensorelements 5L and 5R and derives, from the sensor signals, a differentialsignal that represents the magnetic field as a signal modulation. Thedifferential signal may then be output as an output signal to anexternal controller, control unit or processor (e.g., an electroniccontrol unit (ECU)), or used internally by the sensor circuit forfurther processing (e.g., to generate a pulsed output signal) beforebeing output to an external device. For example, the external device maycount the pulses of the pulsed output signal and calculate a wheel-speedtherefrom.

Alternatively, the differential signal may represent an anglemeasurement, a position measurement, or a rotation direction measurementaccording to the implementation of the sensor device 4 and its targetobject. Furthermore, additional differential pair of sensor elements maybe provided and configured to generate any type of differential signalrepresentative of a differential magnetic field, including those typesof measurements signals provided herein. Also, two or more sets ofdifferential pairs of sensor elements may be used in combination, eachconfigured to generate a differential signal such that additionalinformation related to the target object can be obtained. For example, adifferential speed signal may be used in combination with a differentialdirection signal in an speed sensor. In the this arrangement, the twodifferential sensors may be placed with a certain distance to eachother. As another example, a differential x-angle sensor signal may beused in combination with a differential y-angle sensor signal in anangle sensor.

FIG. 2 is an example of a normalized sinusoidal waveform generated bythe sensors of the sensor device 4 of a magnetic speed sensor. Inparticular, FIG. 2 shows a full revolution speed sensor signal responseof one full revolution of an encoder wheel discussed above in FIG. 1C.However, the signal shape—especially on smaller airgaps—is different anddepends on the shape of the teeth of the wheel.

A pole pair includes an adjacent tooth and notch on a toothed wheel.Typically, for speed applications, a number of teeth on a tooth wheel,translates into a number of sine waveforms for a full revolution of360°. For this example, the encoder wheel 1 would include 24 pole pairsaccording to the sinusoidal waveform shown in FIG. 2.

As can be seen from the waveform, an output signal based on a sensedmagnetic field that oscillates between two extrema (e.g., a minimum anda maximum) in accordance with the rotation of the encoder wheel. Aprocessor may be configured to calculate a wheel-speed and rotationdirection of the rotating encoder wheel 1 based on the output signalsgenerated by the sensors.

FIG. 3 is a schematic block diagram illustrating a magnetic speed sensor300 according to one or more embodiments. The magnetic speed sensor 300includes sensor arrangement X that is configured to generate adifferential sensor signal in response to a magnetic field impingingthereon. In particular, sensor arrangement X may represent a resistorbridge that includes two differential pairs of sensor elements arrangedin a bridge configuration.

The magnetic speed sensor 300 also includes a sensor circuit 30 thatreceives the differential sensor signal from the sensor arrangement Xfor processing and for generation of pulsed output speed signal atoutput 31. In particular, the differential sensor signal may be receivedby an ADC 32 that converts the analog signal into a digital signal, andoutputs the digital signal to a DSP 33 for further processing.

The digital signal processor 33 may include one or more processorsand/or logic units that performs various signal conditioning functions,such as absolute signal conversion, normalization, linearization,frequency increase, and so forth. One or more signal conditioningfunctions may be performed in combination with a lookup table stored inmemory. The output 31 of the digital signal processor 33 may provide oneor more output signals to an external device, such as an ECU. Thedigital signal processor 33 may also be implemented as digital logic.

For example, the speed of rotation of the target object may be output asa speed pulse signal. Thus, the sinusoidal signal generated by thesensor arrangement X may be translated by the digital signal processor33 into pulses, which may be further translated into a movementdetection or a speed output. In addition, the digital signal processor33 may receive two or more differential sensor signals from differentsets of differential pairs of sensor elements for determining additionalinformation related to the target object.

FIG. 4 is a schematic diagram illustrating two possible sensor bridgeconfigurations using four xMR sensor elements. For example, FIG. 4illustrates an example of a magnetic sensor bridge circuit X thatgenerates a differential sensor signal Sx and includes four xMR sensorelements Z1, Z2, Z3, and Z4 with arrows provided to denote a directionof a pinned-layer magnetization of each sensor element aligned in thex-direction. In this case, it can be said that the magnetic sensor hasan x-sensing axis. The sensor is coupled to a magnet that produces astatic magnetic field and that has an MOP of 20 mT.

Sensor elements Z1 and Z2 make up a first pair of sensor elements thatare disposed in a region exposed to a first portion of a differentialmagnetic field having a field strength of −20 mT. Similarly, sensorelements Z3 and Z4 make up a second pair of sensor elements that aredisposed in a region exposed to a second portion of the differentialmagnetic field having a field strength of +20 mT. The first portion andthe second portion of the differential magnetic field have oppositemagnitudes. In addition, sensor element pairs may be placed at locationswhere the field amplitude is different from each other (e.g., at −20 mTand −10 mT).

A first leg of the magnetic sensor bridge circuit X comprises a firstmagnetoresistive sensor element Z1 and a fourth magnetoresistive sensorelement Z4. The first and the fourth magnetoresistive sensor elements Z1and Z4 are connected in series. Furthermore, a second leg of themagnetic sensor bridge circuit X comprises a second magnetoresistivesensor element Z2 and a third magnetoresistive sensor element Z3. Thethird and the second magnetoresistive sensor elements Z3 and Z2 areconnected in series. The first and the third magnetoresistive sensorelements Z1 and Z3 are connected to a first supply terminal of themagnetic sensor bridge circuit X. The second and the fourthmagnetoresistive sensor elements Z2 and Z4 are connected to a second,different supply terminal of the magnetic sensor bridge circuit X. Itwill be appreciated that the specific directions of each pinned-layermagnetization, as shown, may be rotated by 180° as a matter of design.

The differential sensor signal Sx is a function of the magnetic fieldmeasured at the two locations where the two pairs of sensor elements areprovided. As will be described in more detail below, the magnetic sensorbridge circuit X has an asymmetric configuration where the conductanceor the resistance on the left side and the right side of the bridge arenot equal to each other. In this case, the conductance or resistance ofeach sensor element is configured such that the differential sensorsignal Sx is zero when a differential magnetic field at the magneticsensor bridge circuit X equals its MOP field strength.

FIG. 4 further illustrates an example of a magnetic sensor bridgecircuit Y that generates a differential sensor signal Sy and includesfour xMZ sensor elements Z1, Z2, Z3, and Z4 with arrows provided todenote a direction of a pinned-layer magnetization of each sensorelement aligned in the y-direction. In this case, it can be said thatthe magnetic speed sensor 500 has a y-sensing axis.

A first leg of the magnetic sensor bridge circuit Y comprises a firstmagnetoresistive sensor element Z1 and a fourth magnetoresistive sensorelement Z4. The first and the fourth magnetoresistive sensor elements Z1and Z4 are connected in series. Furthermore, a second leg of themagnetic sensor bridge circuit Y comprises a third magnetoresistivesensor element Z3 and a second magnetoresistive sensor element Z2. Thesecond and the third magnetoresistive sensor elements Z2 and Z3 areconnected in series. The first and the third magnetoresistive sensorelements Z1 and Z3 are connected to a first supply terminal of themagnetic sensor bridge circuit Y. The second and the fourthmagnetoresistive sensor elements Z2 and Z4 are connected to a second,different supply terminal of the magnetic sensor bridge circuit Y. Itwill be appreciated that the specific directions of each pinned-layermagnetization, as shown, may be rotated by 180° as a matter of design.

The differential sensor signal Sy is a function of the magnetic fieldmeasured at the two locations where the two pairs of sensor elements areprovided. In this example, with Y-sensitive sensors, the MOP of the leftand right side of the bridge equal to each other (e.g., +20 mT). Thedifferential output Sy is zero.

FIGS. 5A-5C show plan views of different example sensor arrangementconfigurations of a magnetic sensor according to one or moreembodiments. In particular, FIG. 5A shows a chip-layout of a magneticsensor 510, FIG. 5B shows a chip-layout of a magnetic sensor 520, andFIG. 5C shows a chip-layout of a magnetic sensor 530. Each chip-layoutis shown in an x-y plane and includes at least one sensor arrangementhaving an asymmetric bridge configuration.

According to FIG. 5A, the magnetic sensor 510 includes a first sensorarrangement 511 and a second sensor arrangement 512. The first sensorarrangement 511 includes a first pair of sensor elements, includingsensor elements L1 and L2, arranged in a left region (i.e., a left side)of the magnetic sensor 510 and a second pair of sensor elements,including sensor elements R1 and R2, arranged in a right region (i.e., aright side) of the magnetic sensor 510. The first pair of sensorelements L1, L2 and the second pair of sensor elements R1, R2 may beequally spaced from a center of the magnetic sensor 510, which may alsocoincide with a center of a back bias magnet. Therefore, distance d1 mayequal distance d2.

The sensor elements L1, L2, R1, and R2 may be arranged in a bridgecircuit according to one of the examples shown in FIG. 4, and may havecorresponding positions and reference directions similar to those of theresistive elements Z1, Z2, Z3, and Z4, respectively.

The second sensor arrangement 512 may include a monocell sensor elementC. Thus, in a speed sensor application, for example, the first sensorarrangement 511 may be configured to generate a differential speedsignal, and the second sensor arrangement 512 may be configured togenerate a direction signal.

According to FIG. 5B, the magnetic sensor 520 includes a first sensorarrangement 521 and a second sensor arrangement 522. The first sensorarrangement 521 includes a first pair of sensor elements, includingsensor elements L1 and L2, arranged in a first left region (e.g., a leftedge) of the magnetic sensor 520 and a second pair of sensor elements,including sensor elements R1 and R2, arranged in a first right region(e.g., a right edge) of the magnetic sensor 520. The first pair ofsensor elements L1, L2 and the second pair of sensor elements R1, R2 maybe equally spaced from a center of the magnetic sensor 520, which maycoincide with a center of a back bias magnet.

The sensor elements L1, L2, R1, and R2 may be arranged in a bridgecircuit according to one of the examples shown in FIG. 4, and may havecorresponding positions and reference directions similar to those of theresistive elements Z1, Z2, Z3, and Z4, respectively.

The second sensor arrangement 522 includes a third pair of sensorelements, including sensor elements L3 and L4, that may be arranged in asecond left region (e.g., a left center) of the magnetic sensor 520 anda fourth pair of sensor elements, including sensor elements R3 and R4,that may be arranged in a second right region (e.g., a right center) ofthe magnetic sensor 520. The third pair of sensor elements L3, L4 andthe fourth pair of sensor elements R3, R4 may be equally or un-equallyspaced from a center of the magnetic sensor 520, which may coincide witha center of a back bias magnet.

With this, a phase shift between output signals generated by the firstsensor arrangement 521 and the second sensor arrangement 522 is producedand the wheel rotation direction can be detected. The sensor elementsL3, L4, R3, and R4 may be arranged in a bridge circuit according to oneof the examples shown in FIG. 4, and may have corresponding positionsand reference directions similar to those of the resistive elements Z1,Z2, Z3, and Z4, respectively.

In a speed sensor application, for example, the first sensor arrangement521 may be configured to generate a differential speed signal, and thesecond sensor arrangement 522 may be configured to generate adifferential direction signal. With a phase shift between the sensorarrangements 521 and 522, the wheel direction can be determined.

According to FIG. 5C, the magnetic sensor 530 includes a first sensorarrangement 531 and a second sensor arrangement 532. The first sensorarrangement 531 includes a first pair of sensor elements, includingsensor elements L1 and L2, arranged in a first left region (e.g., a leftedge) of the magnetic sensor 520 and a second pair of sensor elements,including sensor elements R1 and R2, arranged in a first right region(e.g., a right center) of the magnetic sensor 520. The first pair ofsensor elements L1, L2 and the second pair of sensor elements R1, R2 maybe offset from a center of the magnetic sensor 530 with respect to eachother. The center of the magnetic sensor 530 may coincide with a centerof a back bias magnet.

The sensor elements L1, L2, R1, and R2 may be arranged in a bridgecircuit according to one of the examples shown in FIG. 4, and may havecorresponding positions and reference directions similar to those of theresistive elements Z1, Z2, Z3, and Z4, respectively.

The second sensor arrangement 532 includes a third pair of sensorelements, including sensor elements L3 and L4, arranged in a second leftregion (e.g., a left center) of the magnetic sensor 530 and a fourthpair of sensor elements, including sensor elements R3 and R4, arrangedin a second right region (e.g., a right edge) of the magnetic sensor520. The third pair of sensor elements L3, L4 and the fourth pair ofsensor elements R3, R4 may be offset from a center of the magneticsensor 530 with respect to each other. The center of the magnetic sensor530 may coincide with a center of a back bias magnet.

The sensor elements L3, L4, R3, and R4 may be arranged in a bridgecircuit according to one of the examples shown in FIG. 4, and may havecorresponding positions and reference directions similar to those of theresistive elements Z1, Z2, Z3, and Z4, respectively.

In a speed sensor application, for example, the first sensor arrangement531 may be configured to generate a differential speed signal, and thesecond sensor arrangement 532 may be configured to generate adifferential direction signal.

FIG. 6 shows a schematic diagram illustrating an asymmetric sensorbridge circuit 600 according to one or more embodiments. The asymmetricsensor bridge circuit includes sensor elements Z1, Z2, Z3, and Z4, assimilarly arranged according to one of the examples shown in FIG. 4. Inaddition, the sensor elements Z1, Z2, Z3, and Z4 may be provided on amagnetic sensor according to any of the sensor arrangements shown inFIGS. 5A-5C. However, it will be appreciated that other sensorarrangements on a magnetic sensor are also possible, and are not limitedthereto.

As can be observed, the sizes of the sensor elements Z1, Z2, Z3, and Z4are exaggerated to illustrate the conductive or resistive asymmetrywithin the bridge circuit 600. For example, an electrical resistance ofa pair of sensor elements on the left side of the asymmetric sensorbridge circuit 600 (i.e., Z1 and Z2) are not equal to an electricalresistance of a pair of sensor elements on the right side of theasymmetric sensor bridge circuit 600 (i.e., Z3 and Z4). In particular,sensor elements Z1 and Z2 may have a larger electrical resistance thanthe electrical resistance of sensor elements Z3 and Z4. Furthermore, theelectrical resistance of sensor elements Z1 and Z2 may be equal orsubstantially equal (e.g., within 5% to allow for manufacturingtolerances), and, similarly, the electrical resistance of sensorelements Z3 and Z4 may be equal or substantially equal (e.g., within 5%to allow for manufacturing tolerances).

The difference in the resistance between the left side of the asymmetricsensor bridge circuit 600 and the right side of the asymmetric sensorbridge circuit 600 is equal to a variation on resistance caused by theMOP field strength of the back bias magnet (e.g., 20 mT). In particular,assuming the MOP is 20 mT, the conductance or resistance of the sensorelements Z1, Z2, Z3, and Z4 is configured such that, if no magneticfield is applied to the asymmetric sensor bridge circuit 600 (i.e., tothe sensor elements Z1, Z2, Z3, and Z4), the differential output voltage(i.e., the differential sensor signal) may be −20 mV. Hence, the −20 mV,in the presence of no magnetic field, represents an electrical DC offsetgenerated by the sensor bridge circuit 600 when coupled with a back biasmagnet that has an MOP field strength of 20 mT.

As the magnetic field also varies over the airgap, the asymmetry of thebridge might also be set to compensate an MOP of 18 mT (and not 20 mT)and result in better offset stability over airgap.

If a differential magnetic field, with a field strength equal to thedifferential MOP field strength (e.g., +/−20 mT) is applied to theasymmetric sensor bridge circuit 600 (i.e., to the sensor elements Z1,Z2, Z3, and Z4), as discussed above, the differential output voltage(i.e., the differential sensor signal) is 0 mV. Thus, an imbalancecaused by the differential MOP field strength is compensated by theelectrical asymmetry of the bridge circuit 600. Due to thiscompensation, no offset cancellation DAC is required and may result is areduction of chip area.

In a similar embodiment, the resistors Z2 and Z4 might have the samesize and the magnetic imbalance is compensated by the imbalance of Z1and Z3.

In a similar embodiment, the resistors Z2 and Z4 might be non-magneticsensitive.

Further, while various embodiments have been described, it will beapparent to those of ordinary skill in the art that many moreembodiments and implementations are possible within the scope of thedisclosure. Accordingly, the invention is not to be restricted except inlight of the attached claims and their equivalents. With regard to thevarious functions performed by the components or structures describedabove (assemblies, devices, circuits, systems, etc.), the terms(including a reference to a “means”) used to describe such componentsare intended to correspond, unless otherwise indicated, to any componentor structure that performs the specified function of the describedcomponent (i.e., that is functionally equivalent), even if notstructurally equivalent to the disclosed structure that performs thefunction in the exemplary implementations of the invention illustratedherein.

Furthermore, the following claims are hereby incorporated into thedetailed description, where each claim may stand on its own as aseparate example embodiment. While each claim may stand on its own as aseparate example embodiment, it is to be noted that—although a dependentclaim may refer in the claims to a specific combination with one or moreother claims—other example embodiments may also include a combination ofthe dependent claim with the subject matter of each other dependent orindependent claim. Such combinations are proposed herein unless it isstated that a specific combination is not intended. Furthermore, it isintended to include also features of a claim to any other independentclaim even if this claim is not directly made dependent to theindependent claim.

It is further to be noted that methods disclosed in the specification orin the claims may be implemented by a device having means for performingeach of the respective acts of these methods.

Further, it is to be understood that the disclosure of multiple acts orfunctions disclosed in the specification or claims may not be construedas to be within the specific order. Therefore, the disclosure ofmultiple acts or functions will not limit these to a particular orderunless such acts or functions are not interchangeable for technicalreasons. Furthermore, in some embodiments a single act may include ormay be broken into multiple sub acts. Such sub acts may be included andpart of the disclosure of this single act unless explicitly excluded.

The techniques described in this disclosure may be implemented, at leastin part, in hardware, software, firmware, or any combination thereof.For example, various aspects of the described techniques may beimplemented within one or more processors, including one or moremicroprocessors, digital signal processors (DSPs), application specificintegrated circuits (ASICs), or any other equivalent integrated ordiscrete logic circuitry, as well as any combinations of suchcomponents. The term “processor” or “processing circuitry” may generallyrefer to any of the foregoing logic circuitry, alone or in combinationwith other logic circuitry, or any other equivalent circuitry. A controlunit including hardware may also perform one or more of the techniquesof this disclosure. Such hardware, software, and firmware may beimplemented within the same device or within separate devices to supportthe various techniques described in this disclosure.

Although various exemplary embodiments have been disclosed, it will beapparent to those skilled in the art that various changes andmodifications can be made which will achieve some of the advantages ofthe concepts disclosed herein without departing from the spirit andscope of the invention. It will be obvious to those reasonably skilledin the art that other components performing the same functions may besuitably substituted. It is to be understood that other embodiments maybe utilized and structural or logical changes may be made withoutdeparting from the scope of the present invention. It should bementioned that features explained with reference to a specific figuremay be combined with features of other figures, even in those notexplicitly mentioned. Such modifications to the general inventiveconcept are intended to be covered by the appended claims and theirlegal equivalents.

What is claimed is:
 1. A magnetic sensor, comprising: a sensorarrangement including a plurality of magnetic field sensor elementselectrically arranged in an asymmetrical bridge circuit, wherein a firsttotal resistance of a first pair of sensor elements provided on a firstside of the asymmetrical bridge circuit is different from a second totalresistance of a second pair of sensor elements provided on a second sideof the asymmetrical bridge circuit, wherein the asymmetrical bridgecircuit is configured to generate a differential signal based on sensorsignals generated by the plurality of magnetic field sensor elements inresponse to a magnetic field impinging thereon, wherein each of thefirst pair of sensor elements has a first resistance value, and each ofthe second pair of sensor elements has a second resistance valuedifferent from the first resistance value.
 2. The magnetic sensor ofclaim 1, wherein: the asymmetric bridge circuit includes a first leg anda second leg connected in parallel between a first supply terminal and asecond supply terminal, wherein both the first leg and the second legextend from the first supply terminal to the second supply terminal, thefirst leg includes a first sensor element of the first pair of sensorelements and a first sensor element of the second pair of sensorelements, and the second leg includes a second sensor element of thefirst pair of sensor elements and a second sensor element of the secondpair of sensor elements.
 3. The magnetic sensor of claim 1, wherein: themagnetic field is a differential magnetic field, and the plurality ofmagnetic field sensor elements include the first pair of sensor elementsconfigured to measure a first portion of the differential magneticfield, and the second pair of sensor elements configured to measure asecond portion of the differential magnetic field, the first portion andthe second portion being different.
 4. The magnetic sensor of claim 3,wherein: the first portion and the second portion have a same fieldstrength with opposing signs.
 5. The magnetic sensor of claim 3,wherein: a difference in resistance between the first pair of sensorelements and the second pair of sensor elements is equal to a variationon the resistance caused by a differential magnetic operation point(MOP) field strength applied to the plurality of magnetic field sensorelements.
 6. The magnetic sensor of claim 3, wherein: each of theplurality of magnetic field sensor elements has a resistive valueconfigured such that the differential signal is zero when a fieldstrength of the differential magnetic field impinging on the pluralityof magnetic field sensor elements is equivalent to a magnetic operationpoint (MOP) field strength.
 7. The magnetic sensor of claim 6, wherein:the resistive value of each of the plurality of magnetic field sensorelements is configured such that the differential signal has an offsetvoltage different than zero when the field strength of the differentialmagnetic field impinging on the plurality of magnetic field sensorelements is zero.
 8. A magnetic sensor, comprising: a sensor arrangementincluding a plurality of magnetic field sensor elements electricallyarranged in an asymmetrical bridge circuit, wherein a first totalresistance of a first pair of sensor elements provided on a first sideof the asymmetrical bridge circuit is different from a second totalresistance of a second pair of sensor elements provided on a second sideof the asymmetrical bridge circuit, wherein the asymmetrical bridgecircuit is configured to generate a differential signal based on sensorsignals generated by the plurality of magnetic field sensor elements inresponse to a magnetic field impinging thereon, wherein the magneticfield is a differential magnetic field, wherein the plurality ofmagnetic field sensor elements include the first pair of sensor elementsconfigured to measure a first portion of the differential magneticfield, and the second pair of sensor elements configured to measure asecond portion of the differential magnetic field, the first portion andthe second portion being different, and wherein: the asymmetric bridgecircuit includes a first leg and a second leg connected in parallelbetween a first supply terminal and a second supply terminal, whereinboth the first leg and the second leg extend from the first supplyterminal to the second supply terminal, the first leg includes: a firstsensor element of the first pair of sensor elements coupled to the firstsupply terminal and having a first resistance value, and a first sensorelement of the second pair of sensor elements coupled to the secondsupply terminal and having a second resistance value, and the second legincludes: a second sensor element of the first pair of sensor elementscoupled to the second supply terminal and having a third resistancevalue different from the second resistance value, and a second sensorelement of the second pair of sensor elements coupled to the firstsupply terminal and having a fourth resistance value different from thefirst resistance value.
 9. A magnetic sensor module comprising: a magnethaving a magnetic operation point (MOP) and configured to produce adifferential magnetic field, the differential magnetic field having adifferential MOP field strength at the MOP, and further having a firstdifferential field portion and a second differential field portion; anasymmetric bridge circuit comprising a plurality of magnetic fieldsensor elements, including a first pair of sensor elements disposed inthe first differential field portion and a second pair of sensorelements disposed in the second differential field portion, wherein afirst total resistance of the first pair of sensor elements provided ona first side of the asymmetrical bridge circuit is different from asecond total resistance of the second pair of sensor elements providedon a second side of the asymmetrical bridge circuit, wherein theasymmetric bridge circuit is configured to generate a differentialsignal based on sensor signals generated by the plurality of magneticfield sensor elements, and wherein the differential signal is zero on acondition that a differential field strength of the differentialmagnetic field impinging on the plurality of magnetic field sensorelements is equivalent to the differential MOP field strength, whereineach of the first pair of sensor elements has a first conductance valueor a first resistance value, and each of the second pair of sensorelements has a second conductance value or a second resistance valuedifferent from the first conductance value and the first resistancevalue, respectively.
 10. The magnetic sensor module of claim 9, wherein:the asymmetric bridge circuit includes a first leg and a second legconnected in parallel between a first supply terminal and a secondsupply terminal, wherein both the first leg and the second leg extendfrom the first supply terminal to the second supply terminal, the firstleg includes a first sensor element of the first pair of sensor elementsand a first sensor element of the second pair of sensor elements, andthe second leg includes a second sensor element of the first pair ofsensor elements and a second sensor element of the second pair of sensorelements.
 11. The magnetic sensor module of claim 9, wherein: adifference in a conductance or a resistance between the first pair ofsensor elements and the second pair of sensor elements is equal to avariation on the capacitance or the resistance caused by thedifferential magnetic operation point (MOP) field strength applied tothe plurality of magnetic field sensor elements.
 12. The magnetic sensormodule of claim 9, wherein the first differential field portion and thesecond differential field portion have a same field strength withopposing signs.
 13. The magnetic sensor module of claim 9, wherein thedifferential signal is a speed signal corresponding to a rotation speedof a target object.
 14. The magnetic sensor module of claim 9, wherein:a magnitude of the differential magnetic field is configured tooscillate between a first extrema and a second extrema in response to arotation of a target object, and the differential signal is one of aspeed signal corresponding to a rotation speed of the target object or adirection signal corresponding to a rotation direction of the targetobject.
 15. The magnetic sensor module of claim 9, wherein the pluralityof magnetic field sensor elements are magnetoresistive sensor elements.16. The magnetic sensor module of claim 9, wherein, on a condition thatthe differential field strength of the differential magnetic fieldapplied to the plurality of magnetic field sensor elements is zero, thedifferential signal has a direct current (DC) offset voltage differentthan zero.
 17. The magnetic sensor of claim 16, wherein the DC offsetvoltage is equal to a peak offset voltage when the differential fieldstrength of the differential magnetic field impinging on the pluralityof magnetic field sensor elements is zero.
 18. A magnetic sensor modulecomprising: a magnet having a magnetic operation point (MOP) andconfigured to produce a differential magnetic field, the differentialmagnetic field having a differential MOP field strength at the MOP, andfurther having a first differential field portion and a seconddifferential field portion; an asymmetric bridge circuit comprising aplurality of magnetic field sensor elements, including a first pair ofsensor elements disposed in the first differential field portion and asecond pair of sensor elements disposed in the second differential fieldportion, wherein the asymmetric bridge circuit is configured to generatea differential signal based on sensor signals generated by the pluralityof magnetic field sensor elements, and wherein the differential signalis zero on a condition that a differential field strength of thedifferential magnetic field impinging on the plurality of magnetic fieldsensor elements is equivalent to the differential MOP field strength,wherein: the asymmetric bridge circuit includes a first leg and a secondleg connected in parallel between a first supply terminal and a secondsupply terminal, wherein both the first leg and the second leg extendfrom the first supply terminal to the second supply terminal, the firstleg includes: a first sensor element of the first pair of sensorelements coupled to the first supply terminal and having a firstconductance value or a first resistance value, and a first sensorelement of the second pair of sensor elements coupled to the secondsupply terminal and having a second conductance value or a secondresistance value, and the second leg includes: a second sensor elementof the first pair of sensor elements coupled to the second supplyterminal and having a third conductance value or a third resistancevalue different from the second conductance value and the secondresistance value, respectively, and a second sensor element of thesecond pair of sensor elements coupled to the first supply terminal andhaving a fourth conductance value or a fourth resistance value differentfrom the first conductance value and the first resistance value,respectively.